Theorem subsnss ≪ | index | src | ≫ 

theorem subsnss (A B: set): $ A C_ B -> subsn B -> subsn A $;
StepHypRefExpression
1 ssel 
A C_ B -> x e. A -> x e. B
2 ssel 
A C_ B -> y e. A -> y e. B
3 2 imim1d 
A C_ B -> (y e. B -> x = y) -> y e. A -> x = y
4 1, 3 imimd 
A C_ B -> (x e. B -> y e. B -> x = y) -> x e. A -> y e. A -> x = y
5 4 alimd 
A C_ B -> A. y (x e. B -> y e. B -> x = y) -> A. y (x e. A -> y e. A -> x = y)
6 5 alimd 
A C_ B -> A. x A. y (x e. B -> y e. B -> x = y) -> A. x A. y (x e. A -> y e. A -> x = y)
7 6 conv subsn 
A C_ B -> subsn B -> subsn A

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_12), axs_set (ax_8)